If we confine ourselves to Lagrange elements for the velocity space, then a reduction of polynomial
degree, while preserving stability, can be done on certain splits (i.e., refinements) of a simplicial mesh.
For example, Zhang proved
stability of the three-dimensional cubic-quadratic pair on Alfeld splits [1]}, and stability
of the three-dimensional quadratic-linear pair on Worsey-Farin splits [2]}.
On the other hand, Zhang suggests that piecewise linear Lagrange elements for the velocity are unstable on Worsey-Farin splits [2]}. One of the main objectives of this paper is to show that indeed the linear case is stable if coupled with the correct pressure space.
